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We suppose that every visitor to a museum, independently from the others, moves around the museum for T minutes, where T is a uniform random variable between 30 and 90 minutes. Moreover, the visitors arrive according to a Poisson process with rate λ = 2 per minute. If the museum opens its doors at 9 a.m. and closes at 6 p.m., find the mean and the variance of the number of visitors in the museum (i) at 10 a.m. and (ii) at time to, where to is comprised between 10:30 a.m. and 6 p.m.

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